A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization

In this paper, we introduce a new approach to develop stochastic optimization algorithms for solving stochastic composite and possibly nonconvex optimization problems. The main idea is to combine two stochastic estimators to form a new hybrid one. We first introduce our hybrid estimator and then investigate its fundamental properties to form a foundation theory for algorithmic development. Next, we apply our theory to develop several variants of stochastic gradient methods to solve both expectation and finite-sum composite optimization problems. Our first algorithm can be viewed as a variant of proximal stochastic gradient methods with a single-loop, but can achieve $\mathcal{O}(\sigma^3\varepsilon^{-1} + \sigma\varepsilon^{-3})$ complexity bound that is significantly better than the $\mathcal{O}(\sigma^2\varepsilon^{-4})$-complexity in state-of-the-art stochastic gradient methods, where $\sigma$ is the variance and $\varepsilon$ is a desired accuracy. Then, we consider two different variants of our method: adaptive step-size and double-loop schemes that have the same theoretical guarantees as in our first algorithm. We also study two mini-batch variants and develop two hybrid SARAH-SVRG algorithms to solve the finite-sum problems. In all cases, we achieve the best-known complexity bounds under standard assumptions. We test our methods on several numerical examples with real datasets and compare them with state-of-the-arts. Our numerical experiments show that the new methods are comparable and, in many cases, outperform their competitors.